Multifrequency electron paramagnetic resonance (EPR) spectroscopy, also called electron spin resonance [1, 2, 3] is a powerful tool for characterising paramagnetic molecules or centres within molecules that contain one or more unpaired electrons. EPR spectra from these systems are often complex and can be interpreted with the aid of a spin Hamiltonian. For an isolated paramagnetic centre (A) a general spin Hamiltonian is [1-3]: EQU H.sub.A =S.multidot.D.multidot.S+.beta.B.multidot.g.multidot.S+S.multidot.A.multid ot.I+I.multidot.Q.multidot.I-.gamma.I.multidot.(l-.sigma.).multidot.B(1)
where S and I are the electron and nuclear spin operators respectively, D is the zero field splitting tensor, g and A are the electron Zeeman and hyperfine coupling matrices respectively, Q is the quadrupole tensor, .gamma. the nuclear gyromagnetic ratio, .sigma. the chemical shift tensor, .beta. the Bohr magneton and B the applied magnetic field.
Additional hyperfine, quadrupole and nuclear Zeeman interactions will be required when superhyperfine splitting is resolved in the experimental EPR spectrum. When two or more paramagnetic centres (A.sub.i, i=1, . . . , N) interact, the EPR spectrum is described by a total spin Hamiltonian (H.sub.Total) which is the sum of the individual spin Hamiltonians (H.sub.A.sbsb.i, Eqn 1) for the isolated centres (A.sub.i) and the interaction Hamiltonian (H.sub.A.sbsb.ij) which accounts for the isotropic exchange, antisymmetric exchange and the anisotropic spin-spin (dipole-dipole coupling) interactions between a pair of paramagnetic centres [2-5]. ##EQU1##
Computer simulation of the experimental randomly orientated or single crystal EPR spectra from isolated or coupled paramagnetic centres is often the only means available for accurately extracting the spin Hamiltonian parameters required for the determination of structural information [2-8]. Computer simulation of randomly orientated EPR spectra is performed in frequency space through the following integration [3,9] ##EQU2## where S(B,.nu..sub.c) denotes the spectral intensity, .vertline..mu..sub.ij .vertline..sup.2 is the transition probability, .nu..sub.c the microwave frequency, .nu..sub.o (B) the resonant frequency, .sigma..sub.v the spectral line width, .function.[.nu..sub.c -.nu..sub.o (B), .sigma..sub.v ] a spectral lineshape function which normally takes the form of either Gaussian or Lorentzian, and C a constant which takes care of all the other experimental parameters. The summation is performed over all the transitions (i,j) contributing to the spectrum and the integration is performed over half of the unit sphere (for ions possessing triclinic symmetry), a consequence of time reversal symmetry [1, 3]. For paramagnetic centres with symmetries higher than triclinic only one or two octants are required.
Unlike most other spectroscopic techniques (nuclear magnetic resonance, infrared and electronic absorption spectroscopy) which are frequency swept, EPR is a field-swept technique. In other words, the resonance condition [h.nu..sub.c =E.sub.i (B.sub.res)-E.sub.j (B.sub.res); where E.sub.i and E.sub.j are the energies (eigenvalues) of the two spin states (eigenvectors) involved in the transition] is achieved by sweeping the magnetic field (varying B.sub.res and hence E.sub.i and E.sub.j).
The methods used for determining B.sub.res in the computer simulation of EPR spectra can be broadly classified into two categories, namely perturbation and matrix diagonalisation methods.
In perturbation methods the energies of the spin states as a function of the field strength B are obtained from analytical expressions. This approach is computationally inexpensive but is limited largely to systems in which a dominant interaction exists and all the other interactions can be treated approximately as perturbations. This approach has been predominantly used to extract spin Hamiltonian parameters from EPR spectra of isolated paramagnetic centres which contain a single unpaired electron (S=1/2) [2, 3, 10-16] and exchange or dipole--dipole coupled binuclear centres (S.sub.A.sbsb.1 =S.sub.A.sbsb.2 =1/2) [2-5].
The second category involves matrix diagonalization and must be employed for calculating the eigenvalues and eigenvectors when perturbation theory breaks down (i.e. two or more interactions have comparable energies). While the eigenvalues are used to calculate B.sub.res, the eigenvectors are used to calculate the transition probability .vertline..mu..sub.ij .vertline..sup.2 [3].
In theory, this approach is general and can be applied to any spin system of choice. However, the numerical integration given in Eqn. 3 can be very time-consuming as the searching process of B.sub.res for a given transition involves substantial computation and this resonance searching process has to be repeated for every transition and for every orientation of the magnetic field. This often involves a very large number (100,000 or more) of matrix diagonalizations. Currently the most efficient algorithms for Hermitian matrix diagonalization are cubic processes O(N.sup.3) (where N is the order of the matrix) [27]. In addition memory requirements can be substantial if the spin space becomes large. Examples of computer simulation software which use this approach to simulate spectra from isolated paramagnetic centres include QPOW [18], EPR.FOR [19], MAGRES [20, 21] and MSPEN/MSGRA [22].
The method described in the international patent application no. PCT/AU96/00534 for computer reconstruction of randomly orientated powder spectra in magnetic resonance, known as the "Sophe" method, features matrix diagonalisation, a segmentation method employing second order eigenfield perturbation theory allowing B.sub.res to be determined quickly, a new scheme for partitioning the unit sphere and the extremely efficient global cubic spline and local linear interpolation schemes for reducing the number of .theta. and .phi. orientations [23-25] while a 26 fold reduction in computational time can be achieved in the simulation of a orthorhombic Cr(III) EPR spectrum using Sophe, even larger reductions can be obtained with larger spin systems. The Sophe computer simulation software package allows the simulation of randomly orientated powder spectra described by either Eqn. 1 or Eqns. 1 and 2 for all spin spin systems (i.e. S.sub.A .gtoreq.1/2 or S.sub.A.sbsb.i .gtoreq.1/2, S.sub.A.sbsb.j .gtoreq.1/2: i, j=1, . . . , N) [23-25] and can easily be extended to other types of randomly orientated powder spectra in magnetic resonance. In the Sophe method [23,24], B.sub.res are calculated by matrix diagonalization only at a number of selected orientations which constitute the vertices of a given Sophe grid, (typically 190 orientations/vertices for orthorhombic symmetry). For all other orientations (normally in the thousands), B.sub.res and .vertline..mu..sub.ij .vertline..sup.2 .vertline..sup.2 are obtained through the Sophe interpolation scheme. Such an approach has been demonstrated to be highly successful for simulating complicated EPR spectra [26-28].
However all computer simulation programmes, including Sophe, that employ matrix diagonalization, are not able to perform the following functions satisfactorily:
(a) tracing of a given transition as a function of orientation in the presence of energy level anti-crossing [29], PA1 (b) tracing of looping transitions [29], PA1 (c) performing the simulations in frequency space [9], and PA1 (d) calculation of the transition probability across a resonant line.
Thus, although the complete matrix diagonalization approach is far superior to perturbation methods, both methods are incapable of tracing the eigenpairs of the matrix to be traced from one field position to another nearby position when there are holes in a transition surface, anti-level crossings, and looping transitions.